chapter 5 dipolar coherence transfer in the presence of chemical
TRANSCRIPT
110
CHAPTER 5
DIPOLAR COHERENCE TRANSFER IN THE PRESENCE OF CHEMICALSHIFT FOR UNORIENTED AND ORIENTED HOMONUCLEAR TWO SPIN-½
SOLID STATE SYSTEMS
Introduction
In this chapter, coherence transfer is examined for the full coupling NMR
Hamiltonian, with the addition of chemical shift. The effects of the full coupling
Hamiltonian are examined for both oriented and unoriented samples using computer
simulation of the full analytical coherence transfer functions. As we have seen in
previous chapters, modeling of dipolar coherence transfer (DCT) for various experiments
and conditions can yield information not easily gained from the end products of
numerical modeling methods. However, the analysis of dipolar coherence transfer, while
not strictly trivial, can seem much more simple when regarding the challenge of
simultaneously modeling all the important interactions present in solid-state NMR. Many
numerical-calculation NMR software packages do very well at modeling pulse sequences
and NMR experiments. Modeling experimental pulse sequences numerically is an
attractive alternative to time-consuming calculations for the analytical expressions of the
Hamiltonian interactions; perhaps so attractive that a complete analysis of the behavior of
two nuclei under dipolar and scalar coupling with a chemical shift addition has not yet
been analyzed extensively in the literature. Equations for all these interactions are readily
111
available [41, 60, 61, 65-68] , but a full analysis was left for calculations in the context of
developing specific experiments.
Solving the analytical expressions for CS , scalar, and dipolar CT relies on two
calculational steps for oriented samples, with an additional third for unoriented (powder)
samples:
Step 1. Calculate analytical expressions for the total interaction
Hamiltonian, for the chemical shift with the scalar and dipolar coupling.
Step 2. Rotationally transform each interaction tensor from its individual
frame of reference into a common (laboratory) observation frame. Calculate the
frequencies of these interactions in this new common frame for their particular values and
angular dependencies, then evaluate these frequencies in the coherence transfer functions.
Step 3. (Unoriented samples) Sum the coherence transfer functions over a
number of crystallite orientations in their arguments to simulate a “powder pattern.”
We find that the steps outlined above can yield complex results. For instance, the
rotational transformation in step 2 can yield some impressively complicated (some might
say Byzantine) trigonometric expressions. In the interests of space, a simplified example
calculation using the chemical shift principal values in the PAS to laboratory
transformation is shown in Figure 5-1 for spin I. The angles (α,β ,γ) reflect the rotational
variables for a the final Wigner rotation from the rotor to the laboratory frame (the other
variables are listed in the caption). Note that Fig. 5-1 is just a deliberate example to show
the typical complexity when facing a problem such as this; the real transform, with all
numerical variables, results in simple frequency components. It is easy to see why
numerical simulations would be widely employed over the analytical….even this
simplified equation took a computer several minutes of processing time.
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Figure 5-1: Example results for the chemical shift frequency A0LAB for spin I in the laboratory frame after
transformation from the principal axis system. The angles relating the CS PAS to the molecular frame wereassigned the values αI =5o, βI = 20 o , γI= 25 o with the values of the chemical shift for spin I being
σI11=46.8ppm, σI
22 = 3.8 ppm, and σI33 = 73.2ppm, with σI = 1/3 (σI
11 + σI22 + σI
33).
The full analytical equation for the equation simplified in Fig. 5-1, with all variables
expressed symbolically, would take up three 8” x 11” sheets of paper. This minute
analysis is not exactly interesting and is time consuming; it yields the right answers but
no new information in itself. However, studying the results of transformations like those
in Figure 5-1 can have limited use. For instance, one can analyze the frequency
dependence on one or more angles. This analysis is necessary for in-depth study of the
coherence transfer with the full coupling Hamiltonian, and will be important for the
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further development of experimental pulse sequences. It will also helpfully act as a solid
roadmap for further work in this direction. In fact, Glaser and Luy have recently
published the coherence transfer functions for three dipolar coupled spins [39] as well as
the superposition of the scalar couplings on the dipolar coherence transfer [125]. These
important papers, coupled with further study of the effects of chemical shift, will help in
the development of experiments to manipulate the Hamiltonians for three or more spins.
Theory for an Oriented Strongly Coupled System
In this chapter, we set out to express the full analytical result for the free evolution
under the full interaction Hamiltonian for a two-spin 1/2 homonuclear system in
uniaxially oriented (or single crystal) samples. We assume that the dipolar coupling as
well as the scalar coupling between the two nuclei is nonzero. Further, relaxation effects
are ignored in the present theoretical study, as we are interested in the dephasing of the
coherence among particular elements of the density matrix describing the system.
The total internal free-evolution Hamiltonian for a two-spin ½ homonuclear
system in the presence of a static external magnetic field consists of chemical shift, scalar
coupling, and dipolar coupling terms:
(5-1)
This Hamiltonian is equivalent to the creation of a zero-field or rotating-frame
Hamiltonian in a solid-state NMR experiment with a non-zero anisotropic chemical shift
offset between two 13C nuclei.
In this work, coherence transfer modes are analyzed under the total coupling
Hamiltonian, HT . HCS can be selectively refocused with π pulses during SSNMR
experiments, but many experiments [7, 20, 57, 91, 93, 96, 114, 127-129] study both
chemical shift and dipolar coupling simultaneously. Chemical shifts can rival HJ and
experimentally recovered HD to maintain a significant contribution to the coherence
114
transfer in solids. Additionally, the relative orientation of the chemical shift anisotropies
to the dipolar frame can make a difference in the behavior of the system. The
superposition of the scalar and dipolar coupling on the chemical shift frequencies are
therefore included in our calculation of the total Hamiltonian.
To begin, the total Hamiltonian, H, for any single interaction can be expressed in
terms of spherical tensors:
∑ ∑=
+
−=−−=
2
0)'()()1(H
k
k
kqqkkq
q tTtA (5-2)
with the spin parameters represented by Tk-q and the spatially-dependent parameters in
the spatial tensor Akq. We are specifically interested in the behavior of the system under
free evolution (no RF); i.e. with the average Hamiltonian, and in the laboratory frame of
reference.
The antisymmetric part of the spin interaction A10 will be neglected because it
does not contribute to the spectrum in first order. In this case, Eq. 5-2 reduces to the two-
term secular expression of the Hamiltonian in spherical coordinates for the chemical shift
in its principal axis system:
20200000 )( TtATAHCS += (5-3a)
which can be cast in the more usual Cartesian basis in the CS principal axis system as:
zz II 0332
21
0 ))(1cos3( ωσσθωσ iiCSH −−+= (5-3b)
The unitary transformation of a spherical tensor is a rotation, and such a rotation
can be achieved by the use of the Wigner rotation formalism. For brevity, we shall just
show the familiar transformation equation from A”2p: to A’20 without addressing the
details, which are present in references such as Mehring [60]:
∑+
−=
−−−=2
2,
)2(0
)2("2
'20 )()(
pqq
iqiqpq
ipp deedeAA θβ ϕγα (5-4)
115
A total transformation from the dipolar PAS to the experimental frame can be
obtained with this equation. Since we wish to include the dipolar coupling with the
chemical shift, this equation alone is not sufficient to put both of these interactions in the
same frame. First, the chemical shift principal axis frame must be rotated into the
molecular (dipolar) frame, and only then can both interactions be simultaneously rotated
into the laboratory frame so that the correct calculations can be made.
The secular part of the dipolar coupling Hamiltonian can be expressed in the form
of Eq. 5-5:
(5-5)
where the second-rank tensors D2q and T2
q define the spatial and spin parts of the dipolar
coupling Hamiltonian (see for example [60]). The secular dipolar coupling Hamiltonian
can be written as
(5-6)
where θ is the angle between the magnetic field and the vector connecting the two spins,
rIS is the distance between I and S nuclei, and γi is the gyromagnetic ratio of nucleus i.
Since the structure of the direct and indirect coupling Hamiltonians for a
homonuclear spin system is similar, the total coupling Hamiltonian is given as
(5-7)
where J is the scalar-coupling constant and DIS is the dipolar coupling frequency is once
again defined as
(5-8)
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In order to evaluate the coherence transfer modes due to the evolution under the coupling
Hamiltonian, we assume a uniaxially oriented system with θ = 0°. Any change in the
value of θ will only change the magnitude of the DIS constant. Equation 5-8 is therefore
the spatial form of the dipolar Hamiltonian in the molecular frame.
The Full Free-Precession Hamiltonian for a strongly coupled oriented sample
The full free-precession Hamiltonian for a strongly coupled, oriented
homonuclear two-spin system in the solid state is a combination of HCS (Eq. 5-3b) and
HJD (Eq. 5-7) in the principal axis system of the dipolar interaction:
SI ⋅−++Ω−Ω−= )2(3^
DJDIzSzSzIzH SI π (5-9)
which can be rewritten for the laboratory reference frame as:
))(2()22()()( 00021
0
^IySyIxSxDJIzSzJDSzIzSzIzH LABLABLABLAB
t +−+++−∆−+Σ−= ππ
(5-10)where D0
LAB is the dipolar constant expressed in laboratory frame in Hz, and J is the
scalar coupling constant in Hz.
The term Σ0LAB is the average frequency of the chemical shifts in the laboratory
frame:
)( ,0
,02
10
SILILAB ωω +=Σ (5-11)
and ∆0LAB is the frequency difference in the laboratory frame:
)( ,0
,00
LSLILAB ωω −=∆ (5-12)
However, since the chemical shift tensor and the dipolar tensor are not assumed to be
colinear, and the chemical shift tensor is not assumed to be axially symmetric, both these
frequencies originate in non-congruent principal axis systems. We must therefore express
the CS frequencies ΩI and ΩS in terms of the dipolar principal axis system. This requires
117
that the chemical shift interaction will be colinear with the dipolar interaction and the
combined frequencies can be transformed identically into the laboratory frame. In
Chapter 2, we have derived the transformation from the dipolar principal axis system to
the laboratory frame. The expression of transformation from the chemical shift principal
axis system to the dipolar principal axis system uses Wigner rotation matrices, ℜ . An
additional transformation from those presented in Chapter 2, that of a rotation from the
chemical shift principal axis system into the dipolar principal axis system, is also
necessary.
A time dependence in the spatial component of the total Hamiltonian will appear
during MAS as a function of the rotor frequency ωr. We assume a time independent
Hamiltonian free of the specific MAS time dependence. We use the secular
approximation when choosing terms from Eq. 5-2 for the total Hamiltonian, that is, we
discard terms of the Hamiltonian with q ≠ 0. This is possible because of stroboscopic
sampling. In MAS, the “first averaging” is taken by sampling at integer multiples of the
Larmor frequency, n2π/ω0, which neglects nonsecular terms. This is valid when ωrÜ ω0.
For 13C in a 600MHz magnet, νC0 =150.96MHz, therefore at a MAS spinning speed of
8kHz, νr/ν0 = 5.3x10-5, which leaves the secular approximation valid under MAS with
proper stroboscopic sampling.
The principal axis system of the chemical shift tensor is the frame in which the
symmetric part of its A2q is diagonal. For the interaction between two homonuclear spins,
the expression for the chemical shift tensor in the principal axis system (PAS) follows
Eq. 5-4 with Akq components for the chemical shift:
31"
00
~
sγTrA = (5-13a)
118
)( 3323"
20 iA σσγ −= (5-13b)
)( 221121"
22 σσγ −=±A (5-13c)
with the isotropic chemical shift is defined as σi= (σ11 + σ22 + σ33 )/3) and γ is the
gyromagnetic ratio. The double hash notation in A”kq indicates that this is the form of the
spatial term in the chemical shift PAS. All other Akq.are non-diagonal and therefore zero
in the PAS.
The observed (laboratory frame) spectral frequency of the chemical shift
interaction is, to first order, determined by the spherical tensor component A’20, which
results from the transformation of the principal axis system, through the goniometer
frame, to the laboratory frame whose z-axis is defined by B0 = Bzz..
The unitary transformation (rotation) of a spherical tensor is expressed as [60]:
∑+
−=
− ℜ≡=k
kp
kpqkqkqkq ARARA )()()( )("1' αβγαβγαβγ (5-14)
where A'kq is the spherical tensor expressed in a new frame (x;, y', z') obtained by this
transformation operator ℜ(αβγ) from the original frame (x,y,z) in which Akp is defined.
With Equation 5-14 we wish to rotate the chemical shift principal axis system to the
molecular (dipolar PAS) frame. Therefore, we must perform three successive rotations,
(that is, three applications of Eq. 5-14) from the chemical shift PAS to the laboratory
frame for both spins I and S independently.
The usual order of operation for the transformation of the molecular (dipolar
PAS) frame to the laboratory (interaction) frame is:
ℜ(ΩMG) ℜ(ΩGL)Dipolar PAS à Goniometer Frame à Laboratory (interaction) frame
119
while for the case of the chemical shift, the order of operations for transformation into the
laboratory frame must contain one additional rotation:
ℜ(ΩPM) ℜ(ΩMG) ℜ(ΩGL)CS PAS à Dipolar PAS à Goniometer Frame à Laboratory (interaction) frame.
where ΩPM = (αX βX γX) are the rotation angles from the PAS of spin X (I or S) to the
molecular frame, ΩMG = (0,β , γ) are the angles from the molecular frame to the
goniometer frame, and ΩGL= (ϕ,θ,0) = (ωrt,θ, 0) are the angles from the goniometer
frame to the laboratory frame, which also is the same as the angles from the rotor frame
(in MAS) to the laboratory frame.
Utilizing the equation 5-14, the three successive rotations can be combined into
one expression for the total rotation from the principal axis system to the laboratory
frame:
120
)()()]()()([ 20
2
2
222
22
,2
2
2
2,0
,0 GLm
mMGqm
CSpmq
CSPMq
PCS
q
CSPMoq
PCSLX AAA ΩℜΩℜΩℜ+Ωℜ+Ωℜ= ∑ ∑−=
−−=
(5-15)
in which we have already expanded the frequency term (An) in the principal axis system
of the chemical shift (CS, P) inside the square brackets.
Each interaction can be rotated from its own principal axis system to the
laboratory frame. Drawing on an earlier example, the transformation of the dipolar
coupling spatial coefficient in Eq. 5-8 to the laboratory frame using the rotation in Eq. 5-
15 results in Eq. 5-16, the expression for the dipolar coefficient in the laboratory frame
(also seen in Chapter 2):
))1cos3)(1cos3(
)cos(2sin2sin
)22cos(sinsin(
2241
43
2243
−−+
+−
+=
θθ
ϕωθθ
ϕωθθ
m
rm
rmISLAB
t
tdD
(5-16)
After transformation of the chemical shift tensor frame to the dipolar (molecular)
frame and then to the laboratory frame, the resulting frequency of spin (I or S) in the
laboratory frame is expressed as ω0X.L, as a sum of terms involving the Euler angles
αXPM, βX
PM , γXPM for spin X (I or S) which transform the CS PAS to the molecular
frame; the angles β and γ, which relate the Euler angle rotations from the molecular
frame (dipolar PAS) to the goniometer frame, and the angles θ and ϕ which relate the
rotation from the rotor frame/goniometer frame to the laboratory frame.
121
Transformation of the Spin Part of HT
Because the sum of the non-secular terms (IxSx + IySy) does not commute with
the difference operator (Iz - Sz), we may not decompose Ht into independent rotations for
each of the operator terms as could be done for HD in Chapters 3 and 4. We must first
find a frame in which the Hamiltonian for these two interactions is diagonal so that we
might apply consecutive rotations. We can then transform the operators into that
reference frame. It so happens that when two operators share the same (diagonal)
Hamiltonian, they commute. We will therefore use the properties of commutation to
bring us into the common frame necessary for calculation.
In order to transform the Hamiltonian into a tilted reference frame in which the
Hamiltonian is diagonal for both the terms (Iz - Sz) and (IxSx + IySy), one must choose
an appropriate axis perpendicular to the two non-commuting operators by which to rotate.
To cause a rotation in a plane (Â, G), where  and G are two cyclically commuting
product operators, an axis C is needed, C = i[Â,G], where C is another product operator.
The problem is solved by transforming Ht into a tilted reference frame, a plane created
between the two axes of (Iz-Sz) and (IxSx + IySy), where the Hamiltonian is diagonal.
This orthogonal operator, (IxSy-IySx), cyclically commutes with (Iz-Sz) and (IxSx+IySy)
and will rotate the Hamiltonian to a tilted reference frame.
Since the two terms in (IxSy-IySx) already commute, the rotation operator for
conversion to the tilted frame is easily defined as a sequential operation (5-17a) and the
rotation back to the laboratory frame is merely the reverse operation (5-17b):
122
ℜT = exp(iθIxSy) exp(-iθIySx). (5-17a)
ℜ†T = exp(iθIySx) exp(-iθIxSy). (5-17b)
The rotation from the lab frame of reference to the tilted frame can be accomplished with
the rotation T = ℜTL ℜ†T and back again into the laboratory frame with the rotation
L = ℜ†T T ℜT .
Now that we have the mechanism to transfer the laboratory-frame operators into a
tilted frame, we must also transform our Hamiltonian into the tilted frame as well. Using
the rotation operator ℜT , HT is rotated into the common frame, resulting in HT’:
AIzSzIzH sIT +Ω−Ω−= ''' (5-18)
with transformed frequencies:
RI 21' +Σ=Ω (5-19a)
RS 21' −Σ=Ω (5-19b)
A = J + 2D (5-19c)
Σ = 1/2(ωILAB + ωS
LAB ) (5-19d)
∆ = ωILAB - ωS
LAB (5-19e)
B = (J - D). (5-19f)
The rotation frame relations
RBsin ? = (5-20a)
R?cos ? = (5-20b)
22 BR +∆= (5-20c)
describe the relation of the frequencies within the original laboratory frame to that of the
new tilted frame. The combined coupling constant B = (J - D), and the chemical shift
123
difference ∆ are the frequencies modulating the original frame's operators; and the
magnitude of the rotated reference frame operator is defined as R in Eq. 5-20c.
In summary, the order of operation on transforming the spin part of the dipolar
Hamiltonian is:
1. Rotate all operators in the laboratory frame to the new, rotated frame with
Eq. 5-17a.
2. Operate on operators thus transformed with the Hamiltonian of
Eq. 5-18.
3. Return the resultant operators back to the laboratory frame with
Eq. 5-17b.
Figure 5-2 . The coordinate system for the rotation to the "tilted frame" that uses the operator (IxSy-IySx),which commutes with both ∆(Iz-Sz) and B(IxSx + IySy). The transformation is achieved by a rotation
angle θ around (IxSy-IySx). See Eq. 5-20 and following for definitions of the constants.
(Iz – Sz)
(IxSx + IySy)
(IxSy-IySx)
θ∆
B
Rotated frame vector
R
124
Methods
Using the methods outlined in the introduction, the resulting operators and their
modulating frequencies can be used to calculate the behavior of a strongly coupled
unoriented system. However, in the strongly coupled oriented system, DIS is dependent
only on rIS assuming θ is equal over all crystallites. In an unoriented system, DIS gains an
angular dependence through the distribution of θIS. Therefore, the characteristic DCT
spectrum of an unoriented system is comprised of the superposition of many frequencies
resulting from the summation over the spatial distribution of DIS. We expect the time-
and frequency-domain components of a sample to reflect the random characteristic of the
spatial anisotropy and produce a spectrum indicative of an unoriented sample; a Pake
doublet (powder pattern) in the case of the frequency-domain, and therefore some kind of
weighted approximate Fourier sum in the case of the time-domain signal (see Fig. 4-2).
Solutions for the unoriented sample FD(t) are calculated numerically over all
space in the analytical functions F’(θ,t). For each initial state σ(0), resulting coherences
evolve, each modulated by frequencies particular to the Hamiltonian. Numerical
simulations of the coherence transfer functions were performed using 250,000 crystallite
orientations with assorted FORTRAN programs performing a Riemann sum over the
analytical coherence transfer functions F’(θ,t), with the angular dependent functions
inside the coherence transfer functions generated by the Wigner transformation in the
program IS.F (see Appendix A). When providing the same results, a smaller number of
crystallites (90,000) are used to speed up computational time.
Results and Discussion
Figure 5-1 gave a glimpse into the complexity of the analytical expressions of the
rotation operations. Even with several parameters expressed numerically, the spatial
Hamiltonian transformation produces a non-trivial collection of terms, much simplified in
125
Figure 5-1. Such a large collection of terms cannot be easily transferred to a computer
program, let alone efficiently powder averaged over the values of the Euler angles α,β ,γ.
A solution is to instead allow the computer program to calculate the frame
transformations from an initial set of PAS values and simultaneously calculate the
powder averaging incrementally over the Euler angles. The program in Appendix A, IS.F,
calculates the PAS è LAB frame transformation for each orientation of the Euler angles
and then sums the frequencies over all space to gain the powder-averaged spectrum of the
full Hamiltonian. The program IS.F uses Wigner matrices to transform the principal
values of the chemical shift and dipolar interactions to a common laboratory frame. To
simulate unoriented sample (powder) patterns, a Riemann sum is executed over the final
Wigner transformation to give the averaging of all crystallites over all space.
Effects of chemical shift coherence transfer in unoriented samples: calculationsfrom the tilted frame.
Using the transformations and rotations in Eqs.5-16 to 5-19, the coefficients after
evolution with the frame-transformed Hamiltonian for the case of σ(0)=Ix are given in
Eq. 5-21 and σ(0)=Iz in Eq. 5-22.
In the coefficients of the Ix operator resulting from evolution of σ(0)=Ix under the
pure dipolar Hamiltonian (see Table 4-1), we have previously seen that the frequency
distribution for an oriented sample with only the coupling Hamiltonian results in a "beat"
phenomenon where two frequencies, 3/2 D and 1/2 D, create a unique coherence transfer
pattern evidenced in Figure 4-1(a) for Equation A in the case of transverse self-
magnetization. However, by including the chemical shift and using the tilted frame
transformation, the resultant coefficient equations for Ix and Sx coherences, are
126
[ ] [ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
+
++
−+
−+
−+
Σ+
=
])sin[ tSsin sin R?]cos[ tScos (sin IySz-
])sin[ tSsin cos R?]cos[ tScos cos (Ix
])cos[ tSsin sin ]sin[ tScos sin R?
( IxSz
])cos[ tSsin cos ]sin[ tScos cos R?
(Iy
]sin[ tSsin sinSy B
]sin[ tSsin cosIzSx B
]sin[ ]sin[ tcosSx B]sin[ tScos cosBIzSy
(t)
2Bt
2At
2Bt
2At
2Bt
2At
2Bt
2At
2Bt
2At
2Bt
2At
2Bt
2At
2Bt
2At
2Bt
2At
2Bt
2At
2Bt
2At
2Bt
2At
RR
RR
xσ
(5-21)
Sz2
]cos[
2
]cos[
IxSy)-)(IySx2R
Bsin[Bt](IySy)(IxSx
2])cos[(
)(
2
222
2
222
R
BtBRIz
R
BtBRR
BtBBt
−∆−+
+∆++
−+∆+∆−
=σ
(5-22)
..]2
cos[]sin[]2
sin[]2
cos[]cos[]2
cos[ toAttBtR
AttBtIxIxTH
+
Σ
∆+Σ→
(5-23a)
..]cos[]2
sin[]2
sin[ totBtAtSxIxTH
+Σ→ (5-23b)
Equation 5-23a and 5-23b reduce correctly for the dipolar Hamiltonian alone, HT=HD,
when J=Σ=∆=0, and in that case the remaining coefficient for Ix is cos(0.5At)cos(0.5Bt),
while that for Sx is sin(0.5At)sin(0.5Bt). These reductions are exactly what is calculated
for evolution under HD (Table 4.1) for functions A and B in Table 4-1.
Likewise, if one sets J=D=0, so that HT = HCS, the coefficient of Ix reduces to
cos(Σt) and that of Sx to cos(Σt) as expected, as both are transverse operators evolving
under a chemical shift term.
127
R term
0
10
20
30
40
D HppmL
-3
-2
-1
0
Dipolar Coupling HkHzL
00.25
0.5
0.75
1
0
10
20
30
40
D HppmL
R term
0
10
20
30
40
D HppmL
-3
-2
-1
0
Dipolar Coupling HkHzL
00.25
0.5
0.75
1
0
10
20
30
40
D HppmL
Figure 5-3 . A graph of the value of R∆
where R=22 B+∆ , as a function of dipolar coupling and ∆,
where (B=2πJ-D).. (a) J=0 Hz. Note the fast shoulder at D=∆=0. (b) J=53Hz. Since J is of opposite sign to
D, term R∆
reaches the value of “1” for larger values of ∆.
5-3b
5-3a
128
0
10
20
30
40
50
S HppmL
0
10
20
30
40
50
D HppmL
-0.5
0
0.5
<Ix>
0
10
20
30
40
50
S HppmL
-0.5
0
0.5
<Ix>
Figure 5-4 : Graph of the “odd term” in Ix for an oriented sample, Eq. , as a function of the chemical shiftaverage frequency in the lab frame (Σ ) and chemical shift difference in the lab frame (∆) where t =1ms,and
the internuclear distance r=1.53Å. The scalar coupling J=0
129
0
20
40
60
D HppmL
-3
-2
-1
0
dip coupling HkHzL
-0.2
0
0.2
0
20
40
60
D HppmL
-0.2
0
0.2
Figure 5-5 : Graph of the“odd term” in Ix for an oriented sample, as a function of the dipolar couplingfrequency in the lab frame in kHz, and the chemical shift difference in the lab frame (∆) where t= 3ms,and
the CS average frequency in the lab frame is Σ=50 ppm. The scalar coupling J=0
130
0
20
40
60
S HppmL
-3
-2
-1
0
dip coupling HkHzL
-1
-0.5
0
0.5
1
0
20
40
60
S HppmL
-1
-0.5
0
0.5
1
Figure 5-6 : Graph of the“odd term” in Ix for an oriented sample, as a function of the dipolar couplingfrequency in the lab frame in kHz, and the chemical shift average in the lab frame (Σ) where t=3ms,and the
CS difference frequency in the lab frame is ∆=50 ppm. The scalar coupling J=0.
131
0
20
40
60
S HppmL
-3
-2
-1
0
dip coupling HkHzL
-1-0.500.51
0
20
40
60
S HppmL
-1-0.500.51
Figure 5-7 : Graph of the“even term” for Ix for an oriented sample, as a function of the dipolar couplingfrequency in the lab frame in kHz, and the chemical shift average in the lab frame (Σ) where t=3ms. The
scalar coupling J=0.
132
0
20
40
60
80
100
S HppmL
-3
-2
-1
0
Dipolar Coupling HkHzL
-0.5
0
0.5
0
20
40
60
80
100
S HppmL
-0.5
0
0.5
Figure 5-8 : Graph of the CT function for Sx when σ(0)=Ix for an oriented sample,, as a function of thedipolar coupling frequency in the lab frame in kHz, and the chemical shift average in the lab frame (Σ )
where t=1ms,and the CS difference frequency in the lab frame is ∆=0 ppm. The scalar coupling J=0. SeeEq. 5-21.
133
0
20
40
60
80
100
S HppmL
-3
-2
-1
0
Dipolar Coupling HkHzL
-0.5
0
0.5
0
20
40
60
80
100
S HppmL
-0.5
0
0.5
Figure 5-9 : Graph of the CT function for Sx when σ(0)=Ix for an oriented sample,, as a function of thedipolar coupling frequency in the lab frame in kHz, and the chemical shift average in the lab frame (Σ )
where t=5ms,and the CS difference frequency in the lab frame is ∆=0 ppm. The scalar coupling J=0. SeeEq. 5-21.
134
0
20
40
60
80
100
S HppmL
-3
-2
-1
0
Dipolar Coupling HkHzL
-0.1
0
0.1
0
20
40
60
80
100
S HppmL
-0.1
0
0.1
Figure 5-10: Graph of the CT function for Sx when σ(0)=Ix for an oriented sample,, as a function of thedipolar coupling frequency in the lab frame in kHz, and the chemical shift average in the lab frame (Σ )
where t=5ms,and the CS difference frequency in the lab frame is ∆=50 ppm. The scalar coupling J=0. SeeEq. 5-21.